Partitioning Hands of Dice

7d6.py

#!/usr/bin/python3

import math
import numpy as np
#import pandas as pd
import matplotlib as mpl
import matplotlib.pyplot as plt

def decrement(X):
	if sum(X) == len(X):	#this is a kludge to make the while loop stop
		X.pop()
		X.append(0)
		return X
	X.sort(reverse=True)
	Y = X.pop()
	if Y > 1:
		Y -= 1
		X.append(Y)
	elif Y == 1:
		decrement(X)
		X.append(X[-1])
	return X

def karmarkar_karp(X):
	Y = list(X)
	while len(Y) > 1:
		Y.sort()
		diff = Y.pop() - Y.pop()
		Y.append(diff)
	return Y[0]

def count_perms(X):
	cts = [0]
	cts *= max(X)
	den = 1
	for i in X:
		cts[i-1] += 1	#i-1 accounts for the zero-indexing
	for i in cts:
		den = den*math.factorial(i)
	num = math.factorial(len(X))
	return num/den

def mechanic(no_dice, no_sides):
	roll = [no_sides]
	roll *= no_dice
	dice = []
	while sum(roll) > (len(roll)-1):
		dice += [list(roll)]
		decrement(roll)
	return dice

#def single_odds(rem_list, tot_list, odd_list):
#	rem = np.array(rem_list)
#	tot = np.array(tot_list)
#	odd = np.array(odd_list)
#	# these arrays are "horizontal"
#	lo_scores = 0.5*(tot - rem)
#	hi_scores = 0.5*(tot + rem)
#	scores = np.hstack((lo_scores, hi_scores))
#	stretch_odds = 0.5*np.hstack((odd,odd))
#	df = pd.DataFrame(np.transpose(np.vstack((scores,stretch_odds))))
#	dist = pd.pivot_table(df, values=1, index=0, aggfunc=np.sum)
#	return dist

m7d6 = mechanic(7,6)

m7d6_sum = []
for roll in m7d6:
	m7d6_sum.append(sum(roll))

m7d6_diff = []
for roll in m7d6:
	m7d6_diff.append(karmarkar_karp(roll))

m7d6_perms = []
for roll in m7d6:
	m7d6_perms.append(count_perms(roll))
m7d6_perms_arr = np.array(m7d6_perms)

m7d6_odds = (1/sum(m7d6_perms_arr))*m7d6_perms_arr

m7d6_xrange = np.array(range(min(m7d6_sum),max(m7d6_sum)))
w, h = mpl.figure.figaspect(0.32)	#aspect found by trial-and-error to look OK
fig1 = plt.figure(figsize=(w,h))
ax11 = plt.hexbin(m7d6_sum, m7d6_diff, gridsize=(17,3), cmap='binary')
#gridsizes are half of ranges, rounded down (one unit of a hexbin is apparently a 4x4 section of hexagons)
plt.xlabel('Maximum Difference (Sum)')
plt.ylabel('Minimum Difference')
plt.box(None)
plt.tick_params(length=0)
#plt.show()
fig1.savefig('figure1.png', bbox_inches='tight')

m7d6_sum_np = np.array(m7d6_sum)
m7d6_diff_np = np.array(m7d6_diff)
m7d6_hi = 0.5*(m7d6_sum_np + m7d6_diff_np)
m7d6_lo = 0.5*(m7d6_sum_np - m7d6_diff_np)
m7d6_half = 0.5*m7d6_sum_np
fig2, ax21 = plt.subplots()
n1, bins1, patches1 = ax21.hist(np.hstack((m7d6_hi, m7d6_lo)), np.arange(2.5,25.5,1),density=1, rwidth=0.5, label='7d6, Partitioned')
n2, bins2, patches2 = ax21.hist(m7d6_half, np.arange(3.25,21.75,0.5), density=1, histtype='step', label='7d6, Halved')
#considered bins1=bins2, but there are also half-points in the second one
plt.xlabel('Individual Score')
plt.box(None)
plt.tick_params(length=0)
ax21.set_yticklabels(['']*len(ax21.get_yticklabels()))
plt.legend(frameon=False)
#plt.show()
fig2.savefig('figure2.png', bbox_inches='tight')

count = [3, 4, 5, 6, 7, 8]	#defining numbers of dice to roll & partition
sides = [4, 6, 8, 10, 12, 20]	#defining the common dice

def chance_diff(no_dice, no_sides):
	rolls = mechanic(no_dice, no_sides)
	rolls_diff = np.array([karmarkar_karp(hand) for hand in rolls], dtype=bool)
#	rolls_diff != 0	#this comparison seems equivalent to "dtype=bool", but causes numerical errors at scale
	rolls_perms = np.array([count_perms(hand) for hand in rolls])
	chance = sum( rolls_diff * rolls_perms / sum(rolls_perms))
	return chance

#def avg_diff(no_dice, no_sides):
#	rolls = mechanic(no_dice, no_sides)
#	#rolls_sum = np.array([sum(hand) for hand in rolls])	#considered "normalizing" by this value, decided against it
#	rolls_diff = np.array([karmarkar_karp(hand) for hand in rolls])
#	rolls_perms = np.array([count_perms(hand) for hand in rolls])
#	avg = sum(rolls_diff * rolls_perms / sum(rolls_perms))
#	return avg

chances = np.array([[chance_diff(a, b) for b in sides] for a in count])

fig3, ax31 = plt.subplots()
im = ax31.imshow(chances, cmap='binary', norm=mpl.colors.Normalize(0,1))
ax31.tick_params(top=True, bottom=False, labeltop=True, labelbottom=False, length=0)
ax31.set_xticks(np.arange(len(sides)))
ax31.set_xticklabels(['d{}'.format(sides[a]) for a in np.arange(len(sides))])
ax31.set_yticks(np.arange(len(count)))
ax31.set_yticklabels(count)
for i in range(len(count)):
	for j in range(len(sides)):
		text = ax31.text(j, i, '{:4.2f}'.format(chances[i, j]), ha='center', va='center', color='w')
for edge, spine in ax31.spines.items():
	spine.set_visible=False
plt.box(None)
#plt.show()
fig3.savefig('figure3.png', bbox_inches='tight')